Lattices of Order-Convex Sets of Forests

Abstract

We give a characterization of the class Co(F ) [Co(F n ), n < ω, respectively] of lattices isomorphic to convexity lattices of posets which are forests [forests of length at most n, respectively], as well as of the class Co(L) of lattices isomorphic to convexity lattices of linearly ordered posets. This characterization yields that the class of finite members from Co(F ) [from Co(F n ), n < ω, or from Co(L)] is finitely axiomatizable within the class of finite lattices.

On Lattices Embeddable into Lattices of Order-Convex Sets. II Star-like Posets

“…In [9], the authors of the present paper prove that SUB(T ) is a finitely based variety, where T denotes the class of posets whose Hasse diagrams are trees, thus generalizing some results from [8]. Moreover, the authors find a characterization of the class of lattices isomorphic to convexity lattices of posets whose Hasse diagrams are forests (trees, totally ordered sets) in [10]. That characterization implies, in particular, that the class of convexity lattices of finite forests (finite trees, finite totally ordered sets, respectively) is finitely axiomatizable within the class Fin of finite lattices.…”